worked on chapter 5

chap_5/e5_6.dfy

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+function Mult(x: nat, y: nat): nat {
+    if y == 0 then 0 else x + Mult(x, y - 1)
+}
+
+lemma {:induction false} MultCommutative(x: nat, y: nat)
+    ensures Mult(x, y) == Mult(y, x)
+{
+    if x == y {
+    } else if x == 0 {
+        MultCommutative(x, y - 1);
+    } else if y < x {
+        MultCommutative(y, x);
+    } else {
+        calc {
+            Mult(x, y);
+        == //Def. Mult
+            x + Mult(x, y - 1);
+        == {MultCommutative(x, y - 1);} 
+            x + Mult(y - 1, x);
+        ==
+            x + y - 1 + Mult(y - 1, x - 1);
+        ==
+            x - 1 + y + Mult(y - 1, x - 1);
+        == {MultCommutative(x - 1, y - 1);}
+            x - 1 + y + Mult(x - 1, y - 1);
+        ==
+            y + Mult(x - 1, y);
+        == {MultCommutative(x - 1, y);}
+            y + Mult(y, x - 1);
+        ==
+            Mult(y, x);
+
+        }
+    }
+}
+
+lemma {:induction false} MultCommutative'(x: nat, y: nat) 
+    ensures Mult(x, y) == Mult(y, x)
+    decreases x + y
+{
+    if x == y {// Trivial
+    } else if x == 0 {
+        MultCommutative'(x, y - 1);
+    } else if y == 0 {
+        MultCommutative'(y, x - 1);
+    } else { // x < y
+        calc {
+            Mult(x, y);
+        == // def. Mult
+            x + Mult(x, y - 1);
+        == {MultCommutative'(y - 1, x);}
+            x + Mult(y - 1, x) ;
+        == // def. Mult
+            x + y - 1 + Mult(y - 1, x - 1);
+        == {MultCommutative'(y - 1, x  -1);}
+            y + x - 1 + Mult(x - 1, y - 1);
+        == // def. Mult
+            y + Mult(x - 1, y);
+        == {MultCommutative'(x - 1, y);}
+            y + Mult(y, x - 1);
+        == // def. Mult
+            Mult(y, x);
+        }
+    }
+}